用有理函数调制高斯小波，提升信号压缩与医学心电检测能力
Rational Gaussian wavelets and corresponding model driven neural networks
- 引入有理函数调节高斯小波形状，自由参数控制波形特征
- 仅用少量小波系数即可逼近复杂信号形态，数值验证有效
- 可嵌入神经网络实现可解释的特征提取，适合医疗信号分析
本文研究基于高斯小波乘以适当有理函数的连续小波变换。该有理函数的零点和极点作为自由参数，显著影响母小波形状，使构造的小波能仅用少数小波系数近似复杂形态信号。我们证明了所提有理高斯小波的适定性，并采用变投影算子进行小波系数的数值逼近。此外，展示了基于变投影的有理高斯小波变换如何用于神经网络，构建高度可解释的特征学习层。通过真实心电图数据中的室性早搏（VEBs）检测任务，验证了该方法的有效性。
原文摘要 · Abstract (English)
In this paper we consider the continuous wavelet transform using Gaussian wavelets multiplied by an appropriate rational term. The zeros and poles of this rational modifier act as free parameters and their choice highly influences the shape of the mother wavelet. This allows the proposed construction to approximate signals with complex morphology using only a few wavelet coefficients. We show that the proposed rational Gaussian wavelets are admissible and provide numerical approximations of the wavelet coefficients using variable projection operators. In addition, we show how the proposed variable projection based rational Gaussian wavelet transform can be used in neural networks to obtain a highly interpretable feature learning layer. We demonstrate the effectiveness of the proposed scheme through a biomedical application, namely, the detection of ventricular ectopic beats (VEBs) in real ECG measurements.
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